Method and an apparatus for use in an electric circuit

ABSTRACT

An apparatus and a method for use in an electric circuit. The apparatus includes a sampling module arranged to obtain an approximation of a voltage-current characteristic of at least one electrical circuit component of the electric circuit subjected to an excitation, wherein each of the at least one electrical circuit component is represented by a constant phase element in an equivalent circuit of the electric circuit; and a processing module arranged to estimate a time-domain voltage response and/or a time-domain current response based on the approximation of the voltage-current characteristic under the excitation applied to the electric circuit.

TECHNICAL FIELD

The present invention relates to a method and an apparatus for use in anelectric circuit, although not exclusively, to a method and an apparatusfor estimating a time-domain response of constant phase elementsubjected to an arbitrary excitation.

BACKGROUND

Electronic or electrical devices usually operate with a predeterminedelectrical characteristic, such as desired current and/or voltageprofile. Each of the electrical components in an electrical device maybe easily analysed using suitable sensors or sampler, however thecharacterization of an electric circuit or network with a number ofelectrical components of different types may be complicated.

Some specific types of electrical components or networks may response indifferently in the time-domain and the frequency-domain. Accordingly,time-domain and frequency-domain response analysis may be useful incharacterizing or predicting the behaviour of the circuit. The analysismay be useful in earlier stages in designing the electronic orelectrical devices.

SUMMARY OF THE INVENTION

In accordance with a first aspect of the present invention, there isprovided a method for use in an electric circuit, comprising the stepsof: obtaining an approximation of a voltage-current characteristic of atleast one electrical circuit component of the electric circuit subjectedto an excitation, wherein each of the at least one electrical circuitcomponent is represented by a constant phase element in an equivalentcircuit of the electric circuit; and estimating a time-domain voltageresponse and/or a time-domain current response based on theapproximation of the voltage-current characteristic under the excitationapplied to the electric circuit.

In an embodiment of the first aspect, the approximation includes acombination of a plurality of discrete-time representations eachrepresents the voltage-current characteristic at a predetermined timeperiod.

In an embodiment of the first aspect, the approximation includes aplurality of voltage response of the at least one electrical circuitcomponent in response to a plurality of current pulses.

In an embodiment of the first aspect, the method further comprises thestep of decomposing a continuous current passing through the at leastone electrical circuit component to the plurality of current pulses.

In an embodiment of the first aspect, each of the plurality of currentpulses are represented by a linear relation.

In an embodiment of the first aspect, the approximation is a zero-orderapproximation.

In an embodiment of the first aspect, the current pulse includes asquare waveform.

In an embodiment of the first aspect, the combination of a plurality ofdiscrete-time representations is expressed as:

${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}{\sum\limits_{k = 0}^{n - 1}{{i_{CPE}\lbrack{kT}\rbrack}\lbrack {( {n - k} )^{\varphi} - ( {n - k - 1} )^{\varphi}} \rbrack}}} + V_{o}}},$

wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period.

In an embodiment of the first aspect, the approximation is a first-orderapproximation.

In an embodiment of the first aspect, the current pulse includes atrapezoidal waveform.

In an embodiment of the first aspect, the current pulse includes aninitial current value and a final current value different from theinitial current value.

In an embodiment of the first aspect, the combination of the pluralityof discrete-time representations is expressed as:

${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{C_{o}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\lbrack {{{i_{CPE}\lbrack{kT}\rbrack}( {n - k} )^{\varphi}} - {{i_{CPE}\lbrack {k + {1T}} \rbrack}( {n - k - 1} )^{\varphi}}} \rbrack} + {\frac{{i_{CPE}\lbrack {k + {1T}} \rbrack} - {i_{CPE}\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi + 2} \rbrack}\lbrack {( {n - k} )^{\varphi + 1} - ( {n - k - 1} )^{\varphi + 1}} \rbrack}} \}}} + V_{o}}},$

wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period.

In an embodiment of the first aspect, each of the at least oneelectrical circuit component exhibits double-layer characteristics.

In an embodiment of the first aspect, each of the at least oneelectrical circuit component includes a battery or a supercapacitor.

In accordance with a second aspect of the present invention, there isprovided an apparatus for use in an electric circuit, comprising asampling module arranged to obtain an approximation of a voltage-currentcharacteristic of at least one electrical circuit component of theelectric circuit subjected to an excitation, wherein each of the atleast one electrical circuit component is represented by a constantphase element in an equivalent circuit of the electric circuit; and aprocessing module arranged to estimate a time-domain voltage responseand/or a time-domain current response based on the approximation of thevoltage-current characteristic under the excitation applied to theelectric circuit.

In an embodiment of the second aspect, the approximation includes acombination of a plurality of discrete-time representations eachrepresents the voltage-current characteristic at a predetermined timeperiod.

In an embodiment of the second aspect, the approximation includes aplurality of voltage response of the at least one electrical circuitcomponent in response to a plurality of current pulses.

In an embodiment of the second aspect, the sampling module is furtherarranged to sample a continuous current passing through the at least oneelectrical circuit component and to decompose the continuous current tothe plurality of current pulses.

In an embodiment of the second aspect, each of the plurality of currentpulses are represented by a linear relation.

In an embodiment of the second aspect, the approximation is a zero-orderapproximation.

In an embodiment of the second aspect, the current pulse includes asquare waveform.

In an embodiment of the second aspect, the combination of a plurality ofdiscrete-time representations is expressed as:

${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}{\sum\limits_{k = 0}^{n - 1}{{i_{CPE}\lbrack{kT}\rbrack}\lbrack {( {n - k} )^{\varphi} - ( {n - k - 1} )^{\varphi}} \rbrack}}} + V_{o}}},$

wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period.

In an embodiment of the second aspect, the approximation is afirst-order approximation.

In an embodiment of the second aspect, the current pulse includes atrapezoidal waveform.

In an embodiment of the second aspect, the current pulse includes aninitial current value and a final current value different from theinitial current value.

In an embodiment of the second aspect, the combination of the pluralityof discrete-time representations is expressed as:

${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{C_{o}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\lbrack {{{i_{CPE}\lbrack{kT}\rbrack}( {n - k} )^{\varphi}} - {{i_{CPE}\lbrack {k + {1T}} \rbrack}( {n - k - 1} )^{\varphi}}} \rbrack} + {\frac{{i_{CPE}\lbrack {k + {1T}} \rbrack} - {i_{CPE}\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi + 2} \rbrack}\lbrack {( {n - k} )^{\varphi + 1} - ( {n - k - 1} )^{\varphi + 1}} \rbrack}} \}}} + V_{o}}},$

wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period.

In an embodiment of the second aspect, each of the at least oneelectrical circuit component exhibits double-layer characteristics.

In an embodiment of the second aspect, each of the at least oneelectrical circuit component includes a battery or a supercapacitor.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will now be described, by way ofexample, with reference to the accompanying drawings in which:

FIGS. 1A to 1C are schematic diagrams of example equivalent circuits andmodeling of a constant phase element (CPE);

FIG. 2 is a schematic diagram of an apparatus for use in an electriccircuit in accordance with an embodiment of the present invention;

FIGS. 3A and 3B are plots showing examples of single current pulses witha square waveform and a trapezoidal waveform respectively;

FIG. 4 is a schematic diagram of an example circuit including a CPE witha parallel resistor;

FIG. 5 is a schematic diagram of an equivalent resistor-capacitor modelof the example circuit of FIG. 4;

FIGS. 6A to 6C are plots showing time domain simulation results of usingzero-order, first-order and RC equivalent approximation of the examplecircuit of FIG. 4 under excitation frequency of 10 mHz, 100 Hz and 1 MHzrespectively;

FIG. 7 is a schematic diagram of an example circuit including aCole-Cole impedance network;

FIGS. 8A to 8D are plots showing sinusoidal current profiles underexcitation frequency of 1 Hz, 10 Hz, 100 Hz and 1 kHz respectively;

FIGS. 9A to 9D are plots showing time domain experimental results of acircuit with a battery in accordance with FIG. 7 and simulation resultsof using zero-order, first-order approximation of the example circuit ofFIG. 7 under excitation frequency of 1 Hz, 10 Hz, 100 Hz and 1 kHzrespectively;

FIG. 10 is a plot showing an arbitrary current profile;

FIG. 11 is a plot showing time domain experimental results of a circuitwith a battery in accordance with FIG. 7 and simulation results of usingzero-order, first-order approximation of the example circuit of FIG. 7under arbitrary current excitation of FIG. 10; and

FIG. 12 is a plot showing a percentage error of the results in FIG. 11between simulation results with the two methods and experiment circuitof FIG. 7.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The inventors have, through their own research, trials and experiments,devised that a constant phase element (CPE) is an electrical circuitcomponent that may be used to model the electrical characteristics of adouble layer as appeared in devices, such as lithium batteries andsupercapacitors.

A double layer or an electrical double-layer may exist on the interfacebetween an electrode and its surrounding electrolyte. It is formed asions from the solution adsorbing onto the electrode surface. The chargedelectrode is separated from the charged ions by an insulating space.Charges separated by an insulator form a capacitor so a bare metalimmersed in an electrolyte will behave like a capacitor.

With reference to FIG. 1A, the impedance of the constant phase element(CPE) Z_(CPE) may be expressed as

$\begin{matrix}{{Z_{CPE}(s)} = \frac{11}{s^{\varphi}C_{o}}} & (1)\end{matrix}$

where s=jω=j(2πf), f is the operating frequency, ϕϵ[0,1] is defined asthe dispersion coefficient and C_(o) is defined as the capacitance ofdouble layer capacitor.

Then, the admittance of Z_(CPE), Y_(CPE), may be expressed as

Y _(CPE)(s)=s ^(ϕ) C _(o)  (2)

By substituting s=jω into (2),

$\begin{matrix}{{Y_{CPE}(\omega)} = {{\omega^{\varphi}C_{o}\cos \frac{\pi}{2}\varphi} + {j\; \omega^{\varphi}C_{o}\sin \frac{\pi}{2}\varphi}}} & (3)\end{matrix}$

Equation (3) can be represented by a parallel RC network, referring toFIG. 1B, with resistor R_(CPE) and capacitor C_(CPE) equal to

$\begin{matrix}{{R_{CPE}( {\varphi,\omega} )} = \frac{1}{\omega^{\varphi}C_{o}\cos \frac{\pi}{2}\varphi}} & (4) \\{{C_{CPE}( {\varphi,\omega} )} = {\omega^{\varphi - 1}C_{o}\sin \frac{\pi}{2}\varphi}} & (5)\end{matrix}$

R_(CPE) and C_(CPE) are dependent on the value of ϕ and the operatingfrequency ω. Some electrical properties are described as follows. TheCPE behaves like a pure resistor for ϕ=0 and a pure capacitor for ϕ=1.The CPE is equivalent to an open circuit for ω=0. Mathematically,

$\begin{matrix}{{R_{CPE}( {0,\omega} )} = \frac{1}{C_{o}}} & (6) \\{{C_{CPE}( {1,\omega} )} = C_{o}} & (7) \\{{R_{CPE}( {\varphi,\omega} )} = \infty} & (8) \\{{C_{CPE}( {\varphi,0} )} = 0} & (9)\end{matrix}$

For ϕ>0, C_(CPE) is nonzero. The initial voltage on C_(CPE) is the sameas the initial voltage V_(CPE,0) across the CPE. As it is a DC value,according to (8), R_(CPE) is infinite, for all ϕ. With reference FIG.1C, the CPE is modelled by a parallel R_(CPE)−C_(CPE) network in serieswith a voltage source V_(o)=V_(CPE,0). For ϕ=0, C_(CPE) is zero, it isunnecessary to consider the initial voltage, as the equivalent circuitof the CPE is a pure resistor. Thus, V_(o)=0.

In some examples, CPE may be considered as an imperfect capacitor. Inorder to describe the electrical characteristics, it is typicallydescribed by measuring its impedance at a particular operatingfrequency. That is, the CPE is driven by a pure sinusoidal voltage orcurrent of predefined magnitude and frequency. Then, the impedance ofthe CPE is calculated by

$\begin{matrix}{{Z_{CPE}( \omega_{o} )} =  {\frac{V_{CPE}}{I_{CPE}}{\angle\theta}} |_{\omega = \omega_{o}}} & (10)\end{matrix}$

where V_(CPE) and I_(CPE) are the magnitudes of v_(CPE) and i_(CPE),respectively, and θ is the phase difference between v_(CPE) and C_(CPE)at the frequency of interest ω_(o).

Based on the impedance characteristics over a range of frequencies, thevalues of ϕ and C_(o) in (1) may be calculated. In some exampleembodiments, complex systems might contain several CPEs. For example,lithium battery may be described by an equivalent electrical circuitwith two, four, or six CPEs, depending on the required level ofaccuracy. Without loss of generality, the values of ϕ and C_(o) of allCPEs in the equivalent circuit are calculated by studying the impedancecharacteristics with electrochemical impedance spectroscopy

Preferably, with the values of ϕ and C_(o), the time-domainvoltage-current characteristics of the CPE operating at a frequency maybe estimated. If the voltage or current excitation contains multiplefrequencies, the time-domain waveforms can be obtained by firstlydecomposing the excitation into multiple frequency components, thendetermining the response of the CPE at the respective operatingfrequency, and finally combining all response. Such method allowsestimating the response of the CPE to arbitrary excitation, it may havesome limitations described as follows.

The frequency spectra of the excitation are practically not known apriori. For example, the discharging current of the batteries on anelectric vehicle is determined by many uncertain factors, such asdriving behaviour of the drivers. Hence, it is necessary to usesophisticated technique, such as Fourier Transform, to calculate thefrequency spectra of the excitation.

The transient response, such as start-up process and large-signaldisturbances, cannot be modelled or described by a spectrum of limitedbandwidth. Thus, the computation burden could be large.

With reference to FIG. 2, there is shown an embodiment of an apparatus100 for use in an electric circuit, comprising a sampling module 102arranged to obtain an approximation of a voltage-current characteristicof at least one electrical circuit component 104 of the electric circuitsubjected to an excitation, wherein each of the at least one electricalcircuit component 104 is represented by a constant phase element 104 inan equivalent circuit of the electric circuit; and a processing module106 arranged to estimate a time-domain voltage response and/or atime-domain current response based on the approximation of thevoltage-current characteristic under the excitation applied to theelectric circuit.

In this embodiment, the sampling module 102 is arranged to obtainelectrical characteristics of one or more electrical circuit component104, for example by measuring the voltage and the current value acrosseach of electrical circuit components 104 of the electrical circuit whenthe electrical circuit are supplied with an electrical current orvoltage excitation. Alternatively, the sampling module 102 may beimplemented to obtain the voltage and current values across theelectrical circuit components 104 in a simulated electrical circuit.

As discussed earlier in this disclosure, a CPE may be used to model theelectrical characteristics of electrical circuit component whichexhibits double layer characteristics in electrical circuits. In somecomplex systems, multiple CPEs may be used to model the electricalcircuit component or a combination of electrical circuit component in anequivalent circuit if such circuit component or a combination of circuitcomponents exhibits double-layer characteristics. Preferably, theequivalent circuit model with multiple CPEs components may be moreaccurate.

In yet an alternative embodiment, the abovementioned electrical circuitcomponent may include one or more CPE in an equivalent circuit of theelectric circuit.

Preferably, the electrical circuit component 104 may be a battery and/ora supercapacitor as discussed in the previous examples. As appreciatedby a person skilled in the art, a battery and/or a supercapacitorexhibit double-layer characteristics in an electric circuit, and maysometimes be referred as an “electric double-layer”. Similarly, thebattery/supercapacitor may be represented by one or more CPE in theequivalent circuit with reference to FIG. 1A.

The processing module 106 is arranged to estimate a time-domain voltageand/or current response based on the approximation of thevoltage-current characteristic, for example by processing each of aplurality of a pulse responses of the electrical circuit components 104of the circuit in the time-domain, such that the time-domain response ofthe electrical circuit may be obtained by combining the analysis of theresponses of the individual components in time-domain. Accordingly, theapproximation may include a combination of a plurality of discrete-timerepresentations each may represent the voltage-current characteristic ata predetermined time period.

Preferably, the approximation may include a plurality of voltageresponses of the at least one electrical circuit component 104 inresponse to a plurality of current pulses. The plurality of currentpulses may be obtained by decomposing a continuous current passingthrough the at least one electrical circuit component 104.Alternatively, the plurality of the current pulses and the correspondingvoltage response may be individually recorded or obtained.

Preferably, the approximation may be a zero-order approximation and/or afirst-order approximation. With reference to FIGS. 3A and 3B, eachcurrent pulse may include a square waveform or a trapezoidal waveform,and both waveforms may be represented by a linear relation.

With reference to FIG. 3A, the CPE may be subjected to a single squarecurrent pulse. The magnitude and duration of the pulse are Î and τ,respectively. The start and end times of the pulse are t_(x) and t_(y)respectively. Thus,

t _(y) =t _(x)+τ  (11)

The current pulse i_(p) can be expressed as

i _(p)(t)=Î{u[t−t _(x)]−u[t−t _(y)]}  (12)

where u is the heaviside step function.

The Laplace-Transformed equation of i_(p) is

$\begin{matrix}{{I_{p}(s)} = {\frac{\hat{I}}{s}( {e^{- {st}_{x}} - e^{- {st}_{y}}} )}} & (13)\end{matrix}$

Thus, based on FIG. 1C, the voltage across the CPE, V_(CPE)(s), may beexpressed as

$\begin{matrix}{\begin{matrix}{{V_{CPE}(s)} = {{{Z_{CPE}(s)}{I_{p}(s)}} + \frac{V_{o}}{s}}} \\{= {{\frac{\hat{I}}{s^{\varphi + 1}C_{o}}( {e^{- {st}_{x}} - e^{- {st}_{y}}} )} + \frac{V_{o}}{s}}}\end{matrix}{where}{V_{o} = \{ {\begin{matrix}V_{{CPE},0} & {{{for}\mspace{14mu} \varphi} > 0} \\0 & {{{for}\mspace{14mu} \varphi} = 0}\end{matrix}.} }} & (14)\end{matrix}$

The inverse Laplace-Transformed equation of (14) is

$\begin{matrix}{{{v_{CPE}(t)} = {{\frac{\hat{I}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}\lbrack {{{u\lbrack {t - t_{x}} \rbrack}( {t - t_{x}} )^{\varphi}} - {{u\lbrack {t - t_{y}} \rbrack}( {t - t_{y}} )^{\varphi}}} \rbrack} + V_{o}}},\mspace{20mu} {{{for}\mspace{14mu} t} > t_{x}}} & (15)\end{matrix}$

where Γ is the Gamma function.

Equation (15) may be rewritten into a discrete-time form as

$\begin{matrix}{{{v_{CPE}({nT})} = {{\frac{\hat{I}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}\lbrack {{{u\lbrack {{nT} - t_{x}} \rbrack}( {{nT} - t_{x}} )^{\varphi}} - {{u\lbrack {{nT} - t_{y}} \rbrack}( {{nT} - t_{y}} )^{\varphi}}} \rbrack} + V_{o}}},\mspace{20mu} {{{for}\mspace{20mu} {nT}} \geq t_{x}}} & (16)\end{matrix}$

where T is the sampling period.

Equation (16) gives the voltage profile of CPE after the CPE is subjectto a single square current pulse. As the CPE current i_(CPE) iscontinuous, it is then decomposed into a series of square pulses at thesampling time instant kT as

$\begin{matrix}{{{i_{CPE}(t)} \approx {\sum\limits_{k = 0}^{n - 1}{{i_{CPE}\lbrack{kT}\rbrack}\{ {{u\lbrack {t - {kT}} \rbrack} - {u\lbrack {t - {( {k + 1} )T}} \rbrack}} \}}}}{{{for}\mspace{14mu} 0} < t \leq {nT}}} & (17)\end{matrix}$

where i_(CPE)[kT] is sampled value of i_(CPE) at t=kT.

Thus, by using (16), the voltage across the CPE can be expressed as thesum of the voltage profiles caused by the series of the current pulsesin (17). Thus, the combination of the plurality of discrete-timerepresentations may be expressed as:

$\begin{matrix}{{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}{\sum\limits_{k = 0}^{n - 1}{{i_{CPE}\lbrack{kT}\rbrack}\lbrack {( {n - k} )^{\varphi} - ( {n - k - 1} )^{\varphi}} \rbrack}}} + V_{o}}} & (18)\end{matrix}$

wherein:ϕϵ[0,1] represents a dispersion coefficient,C_(o) represents a capacitance value of the CPE,Γ represents a Gamma function, andT represents the sampling period.

Alternatively, referring to FIG. 3B, the current pulse may include waveshape which is trapezoidal, i.e. the current pulse includes an initialcurrent value and a final current value different from the initialcurrent value. The time-domain function between the start and end timesof the current pulse is described approximately by a linear equation.

If the current pulse is modeled by a linear function for tϵ[t_(x)t_(y)], i_(p) can be expressed as

$\begin{matrix}{{i_{p}(t)} = {{\{ {{I( t_{x} )} + {\frac{{I( t_{y} )} - {I( t_{x} )}}{\tau}( {t - t_{x}} )}} \} {u\lbrack {t - t_{x}} \rbrack}} - {\{ {{I( t_{y} )} + {\frac{{I( t_{y} )} - {I( t_{x} )}}{\tau}( {t - t_{y}} )}} \} {u\lbrack {t - t_{y}} \rbrack}}}} & (19)\end{matrix}$

where τ=t_(y)−t_(x).

The Laplace-Transformed equation of i_(p) is

$\begin{matrix}{{I_{p}(s)} = {{\lbrack {\frac{I( t_{x} )}{s} + \frac{{I( t_{y} )} - {I( t_{x} )}}{s^{2}\tau}} \rbrack e^{- {st}_{x}}} - {\lbrack {\frac{I( t_{y} )}{s} + \frac{{I( t_{y} )} - {I( t_{x} )}}{s^{2}\tau}} \rbrack e^{- {st}_{y}}}}} & (20)\end{matrix}$

Thus, the voltage across the CPE, V_(CPE)(s), is

$\begin{matrix}{\begin{matrix}{{V_{CPE}(s)} = {{{Z_{CPE}(s)}{I_{p}(s)}} + \frac{V_{o}}{s}}} \\{= {{\frac{1}{s^{\varphi}C_{o}}\begin{bmatrix}{{\{ {\frac{I( t_{x} )}{s} + \frac{{I( t_{y} )} - {I( t_{x} )}}{s^{2}\tau}} \} e^{- {st}_{x}}} -} \\{\{ {\frac{I( t_{y} )}{s} + \frac{{I( t_{y} )} - {I( t_{x} )}}{s^{2}\tau}} \} e^{- {st}_{y}}}\end{bmatrix}} + \frac{V_{o}}{s}}} \\{= {{\frac{1}{C_{o}}\begin{bmatrix}{{\{ {\frac{I( t_{x} )}{s^{\varphi + 1}} + \frac{{I( t_{y} )} - {I( t_{x} )}}{s^{\varphi - 2}\tau}} \} e^{- {st}_{x}}} -} \\{\{ {\frac{I( t_{y} )}{s^{\varphi + 1}} + \frac{{I( t_{y} )} - {I( t_{x} )}}{s^{\varphi + 2}\tau}} \} e^{- {st}_{y}}}\end{bmatrix}} + \frac{V_{o}}{s}}} \\{= {{\frac{1}{C_{o}}\begin{bmatrix}{{\frac{1}{s^{\varphi + 1}}\{ {{{I( t_{x} )}e^{- {st}_{x}}} - {{I( t_{y} )}e^{- {st}_{y}}}} \}} +} \\{\frac{{I( t_{y} )} - {I( t_{x} )}}{s^{\varphi + 2}\tau}\{ {e^{- {st}_{x}} - e^{- {st}_{y}}} \}}\end{bmatrix}} + \frac{V_{o}}{s}}}\end{matrix}{{{where}\mspace{14mu} V_{o}} = \{ {\begin{matrix}V_{{CPE},0} & {{{for}\mspace{14mu} \varphi} > 0} \\0 & {{{for}\mspace{14mu} \varphi} = 0}\end{matrix}.} }} & (21)\end{matrix}$

The Inverse Laplace-Transformed equation of V_(CPE)(s) is

$\begin{matrix}{{{v_{CPE}(t)} = {{\frac{1}{C_{o}}\lbrack {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\{ {{{I( t_{x} )}( {t - t_{x}} )^{\varphi}{u\lbrack {t - t_{x}} \rbrack}} - {{I( t_{y} )}( {t - t_{y}} )^{\varphi}{u\lbrack {t - t_{y}} \rbrack}}} \}} + {\frac{{I( t_{y} )} - {I( t_{x} )}}{{\Gamma \lbrack {\varphi + 2} \rbrack}\tau}\{ {{( {t - t_{x}} )^{\varphi + 1}{u\lbrack {t - t_{x}} \rbrack}} - {( {t - t_{y}} )^{\varphi + 1}{u\lbrack {t - t_{y}} \rbrack}}} \}}} \rbrack} + V_{o}}},{{{for}\mspace{14mu} t} > \tau_{1}}} & (22)\end{matrix}$

where Γ is the Gamma function,

$\begin{matrix}{{{v_{CPE}({nT})} = {{\frac{1}{C_{o}}\lbrack {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\{ {{{I( t_{x} )}( {{nT} - t_{x}} )^{\varphi}{u\lbrack {t - t_{x}} \rbrack}} - {{I( t_{y} )}( {{nT} - t_{y}} )^{\varphi}{u\lbrack {{nT} - t_{y}} \rbrack}}} \}} + {\frac{{I( t_{y} )} - {I( t_{x} )}}{{\Gamma \lbrack {\varphi + 2} \rbrack}\tau}\{ {{( {{nT} - t_{x}} )^{\varphi + 1}{u\lbrack {{nT} - t_{x}} \rbrack}} - {( {{nT} - t_{y}} )^{\varphi + 1}{u\lbrack {{nT} - t_{y}} \rbrack}}} \}}} \rbrack} + V_{o}}},{{{for}\mspace{14mu} {nT}} > \tau_{1}}} & (23)\end{matrix}$

where T is the sampling period.

Equation (23) gives the voltage profile of CPE after the CPE is subjectto a single trapezoidal pulse. As the CPE current C_(CPE) is continuous,it is then decomposed into a series of trapezoidal pulses at thesampling time instant kT as

$\begin{matrix}{{{i_{CPE}(t)} \approx {\sum\limits_{k = 0}^{n - 1}\lbrack {{{i_{CPE}\lbrack{kT}\rbrack}{u\lbrack {t - {kT}} \rbrack}} - {{i_{CPE}\lbrack {k + {1\; T}} \rbrack}{u\lbrack {t - {( {k + 1} )T}} \rbrack}} + {\frac{{i_{CPE}\lbrack {k + {1\; T}} \rbrack} - {i_{CPE}\lbrack{kT}\rbrack}}{\tau}\{ {{( {t - {kT}} ){u\lbrack {t - {kT}} \rbrack}} - {( {t - {( {k + 1} )T}} ){u\lbrack {t - {( {k + 1} )T}} \rbrack}}} \}}} \rbrack}}\mspace{20mu} {{{for}\mspace{14mu} 0} < t \leq {nT}}} & (24)\end{matrix}$

Thus, by using (23), the voltage across the CPE can be expressed as thesum of the voltage profiles caused by the series of the current pulsesin (24). Thus,

$\begin{matrix}{{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{C_{o}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\lbrack {{{i_{CPE}\lbrack{kT}\rbrack}( {n - k} )^{\varphi}} - {{i_{CPE}\lbrack {k + {1\; T}} \rbrack}( {n - k - 1} )^{\varphi}}} \rbrack} + {\frac{{i_{CPE}\lbrack {k + {1\; T}} \rbrack} - {i_{CPE}\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi + 2} \rbrack}\lbrack {( {n - k} )^{\varphi + 1} - ( {n - k - 1} )^{\varphi + 1}} \rbrack}} \}}} + V_{o}}} & (25)\end{matrix}$

wherein:ϕϵ[0,1] represents a dispersion coefficient,C_(o) represents a capacitance value of the CPE,Γ represents a Gamma function, andT represents the sampling period.

Preferably, equations (18) and (25) may use different current pulsemodels to characterize the voltage across a CPE under an arbitrarycurrent through it. These embodiments may be used to characterize thecircuit behaviors and/or operations of more complex circuits, such as aCPE with a parallel resistor and a Cole-Cole impedance network.

With reference to FIG. 4, there is shown a circuit with a CPE inparallel with a resistor R_(ct), which may be considered as a basic unitfor describing the electrical properties of double layer capacitance indevices, such as supercapacitors and lithium batteries.

By using Kirchhoff's voltage and current laws, the voltage across thecircuit, v(t) and current through the circuit i(t) are

$\begin{matrix}{{v(t)} = {v_{CPE}(t)}} & (26) \\\begin{matrix}{{i(t)} = {{i_{CPE}(t)} + {i_{Rct}(t)}}} \\{= {{i_{CPE}(t)} + \frac{v(t)}{R_{ct}}}}\end{matrix} & (27)\end{matrix}$

Based on (27),

$\begin{matrix}{{i_{CPE}(t)} = {{i(t)} - \frac{v(t)}{R_{ct}}}} & (28)\end{matrix}$

By using (26), (28), and (18), v(nT) can be expressed as

$\begin{matrix}{{v({nT})} = {{\frac{T^{\varphi}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}{\sum\limits_{k = 0}^{n - 1}{\{ {{i\lbrack{kT}\rbrack} - \frac{v\lbrack{kT}\rbrack}{R_{ct}}} \} \lbrack {( {n - k} )^{\varphi} - ( {n - k - 1} )^{\varphi}} \rbrack}}} - V_{o}}} & (29)\end{matrix}$

Equation (29) gives the voltage-current characteristics of the module inFIG. 4 with the sampled current modeled by square pulses shown in FIG.3A.

By using (26), (28), and (25), v(nT) can be expressed as

$\begin{matrix}{{v({nT})} = {{\frac{T^{\varphi}}{C_{o}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\lbrack {{\{ {{i\lbrack{kT}\rbrack} - \frac{v_{CPE}\lbrack{kT}\rbrack}{R_{ct}}} \} ( {n - k} )^{\varphi}} - {\{ {{i\lbrack {k + {1\; T}} \rbrack} - \frac{v_{CPE}\lbrack{kT}\rbrack}{R_{ct}}} \} ( {n - k - 1} )^{\varphi}}} \rbrack} + {\frac{{i\lbrack {k + {1\; T}} \rbrack} - {i\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi + 2} \rbrack}\lbrack {( {n - k} )^{\varphi + 1} - ( {n - k - 1} )^{\varphi + 1}} \rbrack}} \}}} + V_{o}}} & (30)\end{matrix}$

Equation (30) gives the voltage-current characteristics of the module inFIG. 4 with the sampled current modeled by trapezoidal pulses shown inFIG. 3B.

The voltage-current characteristics given in (29) and (30) are evaluatedand compared with the resistor-capacitor (R_(eq)−C_(eq)) model shown inFIG. 5, in which the values of the resistor R_(eq) and capacitor C_(eq)are frequency dependent. The CPE has ϕ=0.7538 and C_(o)=53.54 [Eq. (1)].Assume that the initial voltage V_(o) is zero. The testing frequenciesinclude 10 mHz, 100 Hz, and 1 MHz. The input current is sinusoidal withmagnitude of 1 A. Table I shows the equivalent values of resistor R_(eq)and capacitor C_(eq) at the considered frequencies.

TABLE I Values of R_(eq) and C_(eq) at different frequencies FrequencyR_(CPE) C_(CPE) R_(ct) R_(eq) C_(eq) 10 mHz 0.3988 98.0046 0.3403 0.183698.0046 100 Hz 3.8506e−4 10.1495 0.3403 3.8462e−4 10.1495 1 MHz3.7181e−4 1.0511 0.3403 3.7181e−7 1.0511

Table II shows the steady-state root-mean-square value of voltage v(t)and phase difference between v(t) and i(t) obtained by (29), (30), andequivalent R_(eq)−C_(eq) model when the excitation frequency is 10 mHz.Table III shows the results when the excitation frequency is 100 Hz.Table IV shows the results when the excitation frequency is 1 MHz. Thevalue of T in (29) and (30) is 27.78 μs. It can be observed from theresults that the voltages obtained by the three methods are close. Withreference to FIGS. 6A to 6C, there are shown simulation results with thethree methods under three different frequencies.

TABLE II With excitation frequency of 10 mHz Model RMS Voltage (V) Phase(deg) Equation (29) 1.7225e−1 48.4517 Equation (30) 1.7225e−1 48.6992R_(eq)-C_(eq) model 1.7197e−1 48.5080

TABLE III With excitation frequency of 100 Hz Model RMS Voltage (V)Phase (deg) Equation (29) 2.0538e−4 68.0695 Equation (30) 2.0536e−467.8220 R_(eq)-C_(eq) model 2.0536e−4 67.8188

TABLE IV With excitation frequency of 1 MHz Model RMS Voltage (V) Phase(deg) Equation (29) 1.9832e−7 68.0921 Equation (30) 1.9834e−7 67.8446R_(eq)-C_(eq) model 1.9832e−7 67.8414

These results obtained by equivalent R_(eq)−C_(eq) model are taken asreference, and Table V shows the discrepancies of the results obtainedby equations (29) and (30).

TABLE V Discrepancies of the methods with equations (29) and (30) Errorof RMS Voltage (%) Error of Phase (%) Frequency With (29) With (30) With(29) With (30) 10 mHz 1.6447e−1 1.6447e−1 3.9416e−1 1.1606e−1 100 Hz 01.0330e−2 3.6966e−1 4.7185e−3 1 MHz 3.5656e−3 1.4262e−2 3.6954e−14.7169e−3

The results as illustrated in the Figures and the above tables showsthat the values obtained using these different methods are very close,therefore the plots of the values obtained by different methodssubstantially overlap with each other.

With reference to FIG. 7, there is shown an example Cole-Cole impedancenetwork with two CPEs. It consists of an output resistor R_(o), and twoCPE-R_(ct) circuits. Such network is popularly used to characterize thestatic and dynamic behaviors of batteries.

By using Kirchhoff's voltage and current laws, the voltage across thecircuit v(t) and current through the circuit i(t) can be expressed as

$\begin{matrix}{{v(t)} = {{{i(t)}R_{o}} + {v_{{CPE},1}(t)} + {v_{{CPE},2}(t)}}} & (31) \\\begin{matrix}{{i(t)} = {{i_{{CPE},1}(t)} + {i_{{Rct},1}(t)}}} \\{= {{i_{{CPE},1}(t)} + \frac{v_{{CPE},1}(t)}{R_{{ct},1}}}}\end{matrix} & (32) \\\begin{matrix}{{i(t)} = {{i_{{CPE},2}(t)} + {i_{{Rct},2}(t)}}} \\{= {{i_{{CPE},2}(t)} + \frac{v_{{CPE},2}(t)}{R_{{ct},2}}}}\end{matrix} & (33)\end{matrix}$

Based on (32),

$\begin{matrix}{{i_{{CPE},1}(t)} = {{i(t)} - \frac{v_{{CPE},1}(t)}{R_{{ct},1}}}} & (34)\end{matrix}$

Based on (33),

$\begin{matrix}{{i_{{CPE},2}(t)} = {{i(t)} - \frac{v_{{CPE},2}(t)}{R_{{ct},2}}}} & (35)\end{matrix}$

By using (34), (35), and (18), v_(CPE,1)(nT) and v_(CPE,2)(nT) can beexpressed as

$\begin{matrix}{{v_{{CPE},1}({nT})} = {{\frac{T^{\varphi_{1}}}{{\Gamma \lbrack {\varphi_{1} + 1} \rbrack}C_{o,1}}{\sum\limits_{k = 0}^{n - 1}{\{ {{i\lbrack{kT}\rbrack} - \frac{v_{{CPE},1}\lbrack{kT}\rbrack}{R_{{ct},1}}} \} \lbrack {( {n - k} )^{\varphi_{1}} - ( {n - k - 1} )^{\varphi_{1}}} \rbrack}}} + V_{o,1}}} & (36) \\{{v_{{CPE},2}({nT})} = {{\frac{T^{\varphi_{2}}}{{\Gamma \lbrack {\varphi_{2} + 1} \rbrack}C_{o,2}}{\sum\limits_{k = 0}^{n - 1}{\{ {{i\lbrack{kT}\rbrack} - \frac{v_{{CPE},2}\lbrack{kT}\rbrack}{R_{{ct},2}}} \} \lbrack {( {n - k} )^{\varphi_{2}} - ( {n - k - 1} )^{\varphi_{2}}} \rbrack}}} + V_{o,2}}} & (37)\end{matrix}$

where ϕ₁ and ϕ₂ are the dispersion coefficient of CPE1 and CPE2,respectively, C_(o,1) and C_(o,2) are the double layer capacitance ofCPE1 and CPE2, respectively, and V_(o,1) and V_(o,2) are the initialvoltages across CPE1 and CPE2, respectively.

Equations (36) and (37) give the voltage-current characteristics of thetwo CPE-Rct circuits in FIG. 7 with the sampled current modeled bysquare pulses shown in FIG. 3A. The voltage-current characteristic ofthe impedance network is obtained by substituting (36) and (37) into(31).

By using (34), (35), and (25), v_(CPE,1)(nT) and v_(CPE,2)(nT) can beexpressed as

$\begin{matrix}{{v_{{CPE},1}({nT})} = {{\frac{T^{\varphi_{1}}}{C_{o,1}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi_{1} + 1} \rbrack}\lbrack {{\{ {{i\lbrack{kT}\rbrack} - \frac{v_{{CPE},1}\lbrack{kT}\rbrack}{R_{{ct},1}}} \} ( {n - k} )^{\varphi_{1}}} - {\{ {{i\lbrack {k + {1\; T}} \rbrack} - \frac{v_{{CPE},1}\lbrack{kT}\rbrack}{R_{{ct},1}}} \} ( {n - k - 1} )^{\varphi_{1}}}} \rbrack} + {\frac{{i\lbrack {k + {1\; T}} \rbrack} - {i\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi_{1} + 2} \rbrack}\lbrack {( {n - k} )^{\varphi_{i} + 1} - ( {n - k - 1} )^{\varphi_{1} + 1}} \rbrack}} \}}} + V_{o,1}}} & (38) \\{{v_{{CPE},2}({nT})} = {{\frac{T^{\varphi_{2}}}{C_{o,2}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi_{2} + 1} \rbrack}\lbrack {{\{ {{i\lbrack{kT}\rbrack} - \frac{v_{{CPE},2}\lbrack{kT}\rbrack}{R_{{ct},2}}} \} ( {n - k} )^{\varphi_{2}}} - {\{ {{i\lbrack {k + {1\; T}} \rbrack} - \frac{v_{{CPE},2}\lbrack{kT}\rbrack}{R_{{ct},2}}} \} ( {n - k - 1} )^{\varphi_{2}}}} \rbrack} + {\frac{{i\lbrack {k + {1\; T}} \rbrack} - {i\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi_{2} + 2} \rbrack}\lbrack {( {n - k} )^{\varphi_{2} + 1} - ( {n - k - 1} )^{\varphi_{2} + 1}} \rbrack}} \}}} + V_{o,2}}} & (39)\end{matrix}$

Equations (38) and (39) give the voltage-current characteristics of thetwo CPE-Rct circuits in FIG. 7 with the sampled current modeled bytrapezoidal pulses shown in FIG. 3B. The voltage-current characteristicof the impedance network is obtained by substituting (38) and (39) into(31).

The voltage-current characteristics given in (38) and (39) are evaluatedand compared with the experimental results. In this example, a Li-ionbattery which has 130 mAH capacity has been used for the experimentalverifications. Table VI shows the corresponding components value inCole-Cole impedance network. Assume that the initial voltage V_(o,1) andV_(o,2) are zero. The experiments are conducted with sinusoidal andarbitrary profile.

For the sinusoidal current, the testing frequencies include 1 Hz, 10 Hz,100 Hz, and 1 kHz with magnitude of 100 mA and a DC offset 1 A. FIGS. 8Ato 8D show the current profiles under four different frequencies.

TABLE VI Components values of Cole-Cole impedance network in evaluationR_(o) ϕ₁ C_(o, 1) R_(ct, 1) ϕ₂ C_(o, 2) R_(ct, 2) 0.1103 0.7538 53.540.3403 0.4114 0.4554 0.0990

Table VII shows the steady-state root-mean-square value of voltage v(t)and phase difference between v(t) and i(t) obtained by (38), (39), andexperimental results when the excitation frequency is 1 Hz. Table VIIIshows the results when the excitation frequency is 10 Hz. Table IX showsthe results when the excitation frequency is 100 Hz. Table X shows theresults when the excitation frequency is 1 kHz. The value of T in (38)and (39) is 10 ms. It can be observed from the results that the voltagesobtained by the two methods are close to the experiments. FIGS. 9A to 9Dshow the simulation results with the two methods under four differentfrequencies.

TABLE VII With excitation frequency of 1 Hz Model RMS Voltage (V) Phase(deg) Equation (38) 3.0389 26.1837 Equation (39) 3.0389 26.1837Experimental 3.0780 28.1231

TABLE VIII With excitation frequency of 10 Hz Model RMS Voltage (V)Phase (deg) Equation (38) 2.9894 157.3685 Equation (39) 2.9894 157.3697Experimental 3.0312 158.0024

TABLE IX With excitation frequency of 100 Hz Model RMS Voltage (V) Phase(deg) Equation (38) 2.9417 95.9637 Equation (39) 2.9417 95.9646Experimental 2.9823 98.6328

TABLE X With excitation frequency of 1 kHz Model RMS Voltage (V) Phase(deg) Equation (38) 2.8965 35.9279 Equation (39) 2.8965 35.9282Experimental 2.9340 35.4900

If the results obtained by experiment are taken as reference, Table XIshows the discrepancies of the results obtained by equations (38) and(39).

TABLE XI Discrepancies of the methods with equations (38) and (39) Errorof RMS Voltage (%) Error of Phase (%) Frequency With (38) With (39) With(38) With (39) 1 Hz −1.27031 −1.27031 −6.8961 −6.8961 10 Hz −1.37899−1.37899 −0.4012 −0.4004 100 Hz −1.36137 −1.36137 −2.7061 −2.7052 1 kHz−1.27812 −1.27812 1.23386 1.2347

The results as illustrated in the Figures and the above tables showsthat the values obtained using square pulse approximation and thetrapezoidal pulse approximation methods are very close, therefore theplots of the values obtained by these two methods substantially overlapwith each other.

With reference to FIG. 10, there is shown the arbitrary current profile.Table XII shows the steady-state root-mean-square value of voltage v(t)and phase difference between v(t) and i(t) obtained by (38), (39), andexperimental results with the arbitrary current profile. The value of Tin (38) and (39) is 10 ms. It can be observed from the results that thevoltages obtained by the two methods are close to the experiments. FIG.11 shows the simulation results with the two methods under the arbitrarycurrent profile excitation.

TABLE XII With arbitrary current profile Model RMS Voltage (V) Phase(deg) Equation (38) 3.1617 45.1196 Equation (39) 3.1617 45.1189Experimental 3.2018 45.2458

If the results obtained by experiment are taken as reference, Table XIIIshows the discrepancies of the results obtained by equations (38) and(39). FIG. 12 shows the percentage error between two methods andexperiment.

TABLE XIII Discrepancies of the methods with equations (38) and (39)Error of RMS Voltage (%) Error of Phase (%) Max Error (%) With With WithWith With With (38) (39) (38) (39) (38) (39) −1.25242 −1.25242 −0.2789−0.2789 −0.0258 −0.0287

Similar to the previous examples, the results as illustrated in theFIGS. 11 and 12 and the above tables shows that the values obtainedusing square pulse approximation and the trapezoidal pulse approximationmethods are very close, therefore the plots of the values obtained bythese two methods substantially overlap with each other.

These embodiments may be advantageous in that the method allowsdetermining the time-domain response of the CPE to arbitrary excitationwithout dealing with the frequency response of the CPE. By applyingzero- or first-order approximation to the waveforms, well-defineddiscrete-time functions for describing the relationships between thevoltage and current of the CPE are derived.

Advantageously, the method and apparatus is generic and may be extendedreadily to estimate the time-domain voltage and current waveforms ofelectrical circuits with multiple CPEs and other circuit elements underarbitrary excitation. In addition, the method and apparatus may beapplied in both simulated and real electric circuit.

It will be appreciated by persons skilled in the art that numerousvariations and/or modifications may be made to the invention as shown inthe specific embodiments without departing from the spirit or scope ofthe invention as broadly described. The present embodiments are,therefore, to be considered in all respects as illustrative and notrestrictive.

It will also be appreciated that where the methods and systems of thepresent invention may be either wholly implemented by computing systemor partly implemented by computing systems then any appropriatecomputing system architecture may be utilised. This will includestandalone computers, network computers and dedicated hardware devices.Where the terms “computing system” and “computing device” are used,these terms are intended to cover any appropriate arrangement ofcomputer hardware capable of implementing the function described.

Any reference to prior art contained herein is not to be taken as anadmission that the information is common general knowledge, unlessotherwise indicated.

1. A method for use in an electric circuit, comprising the steps of:obtaining an approximation of a voltage-current characteristic of atleast one electrical circuit component of the electric circuit subjectedto an excitation, wherein each of the at least one electrical circuitcomponent is represented by a constant phase element in an equivalentcircuit of the electric circuit; and estimating a time-domain voltageresponse and/or a time-domain current response based on theapproximation of the voltage-current characteristic under the excitationapplied to the electric circuit.
 2. A method in accordance with claim 1,wherein the approximation includes a combination of a plurality ofdiscrete-time representations each represents the voltage-currentcharacteristic at a predetermined time period.
 3. A method in accordancewith claim 2, wherein the approximation includes a plurality of voltageresponse of the at least one electrical circuit component in response toa plurality of current pulses.
 4. A method in accordance with claim 3,further comprising the step of decomposing a continuous current passingthrough the at least one electrical circuit component to the pluralityof current pulses.
 5. A method in accordance with claim 4, wherein eachof the plurality of current pulses are represented by a linear relation.6. A method in accordance with claim 3, wherein the approximation is azero-order approximation.
 7. A method in accordance with claim 3,wherein the current pulse includes a square waveform.
 8. A method inaccordance with claim 6, wherein the combination of a plurality ofdiscrete-time representations is expressed as:${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}{\sum\limits_{k = 0}^{n - 1}{{i_{CPE}\lbrack{kT}\rbrack}\lbrack {( {n - k} )^{\varphi} - ( {n - k - 1} )^{\varphi}} \rbrack}}} + V_{o}}},$wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period.
 9. Amethod in accordance with claim 3, wherein the approximation is afirst-order approximation.
 10. A method in accordance with claim 3,wherein the current pulse includes a trapezoidal waveform.
 11. A methodin accordance with claim 3, wherein the current pulse includes aninitial current value and a final current value different from theinitial current value.
 12. A method in accordance with claim 9, whereinthe combination of the plurality of discrete-time representations isexpressed as:${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{C_{o}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\lbrack {{{i_{CPE}\lbrack{kT}\rbrack}( {n - k} )^{\varphi}} - {{i_{CPE}\lbrack {k + {1\; T}} \rbrack}( {n - k - 1} )^{\varphi}}} \rbrack} + {\frac{{i_{CPE}\lbrack {k + {1\; T}} \rbrack} - {i_{CPE}\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi + 2} \rbrack}\lbrack {( {n - k} )^{\varphi + 1} - ( {n - k - 1} )^{\varphi + 1}} \rbrack}} \}}} + V_{o}}},$wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period.
 13. Amethod in accordance with claim 1, wherein each of the at least oneelectrical circuit component is arranged to exhibit double-layercharacteristics.
 14. A method in accordance with claim 1, wherein eachof the at least one electrical circuit component includes a battery or asupercapacitor.
 15. An apparatus for use in an electric circuit,comprising: a sampling module arranged to obtain an approximation of avoltage-current characteristic of at least one electrical circuitcomponent of the electric circuit subjected to an excitation, whereineach of the at least one electrical circuit component is represented bya constant phase element in an equivalent circuit of the electriccircuit; and a processing module arranged to estimate a time-domainvoltage response and/or a time-domain current response based on theapproximation of the voltage-current characteristic under the excitationapplied to the electric circuit.
 16. An apparatus in accordance withclaim 15, wherein the approximation includes a combination of aplurality of discrete-time representations each represents thevoltage-current characteristic at a predetermined time period.
 17. Anapparatus in accordance with claim 16, wherein the approximationincludes a plurality of voltage response of the at least one electricalcircuit component in response to a plurality of current pulses.
 18. Anapparatus in accordance with claim 17, wherein the sampling module isfurther arranged to sample a continuous current passing through the atleast one electrical circuit component and to decompose the continuouscurrent to the plurality of current pulses.
 19. An apparatus inaccordance with claim 18, wherein each of the plurality of currentpulses are represented by a linear relation.
 20. An apparatus inaccordance with claim 17, wherein the approximation is a zero-orderapproximation.
 21. An apparatus in accordance with claim 17, wherein thecurrent pulse includes a square waveform.
 22. An apparatus in accordancewith claim 20, wherein the combination of the plurality of discrete-timerepresentations is expressed as:${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{{\Gamma \lbrack {\varphi + 1} \rbrack}C_{o}}{\sum\limits_{k = 0}^{n - 1}{{i_{CPE}\lbrack{kT}\rbrack}\lbrack {( {n - k} )^{\varphi} - ( {n - k - 1} )^{\varphi}} \rbrack}}} + V_{o}}},$wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period. 23.An apparatus in accordance with claim 17, wherein the approximation is afirst-order approximation.
 24. An apparatus in accordance with claim 17,wherein the current pulse includes a trapezoidal waveform.
 25. Anapparatus in accordance with claim 17, wherein the current pulseincludes an initial current value and a final current value differentfrom the initial current value.
 26. An apparatus in accordance withclaim 23, wherein the combination of the plurality of discrete-timerepresentations is expressed as:${{v_{CPE}({nT})} = {{\frac{T^{\varphi}}{C_{o}}{\sum\limits_{k = 0}^{n - 1}\{ {{\frac{1}{\Gamma \lbrack {\varphi + 1} \rbrack}\lbrack {{{i_{CPE}\lbrack{kT}\rbrack}( {n - k} )^{\varphi}} - {{i_{CPE}\lbrack {k + {1\; T}} \rbrack}( {n - k - 1} )^{\varphi}}} \rbrack} + {\frac{{i_{CPE}\lbrack {k + {1\; T}} \rbrack} - {i_{CPE}\lbrack{kT}\rbrack}}{\Gamma \lbrack {\varphi + 2} \rbrack}\lbrack {( {n - k} )^{\varphi + 1} - ( {n - k - 1} )^{\varphi + 1}} \rbrack}} \}}} + V_{o}}},$wherein: ϕϵ[0,1] represents a dispersion coefficient, C_(o) represents acapacitance value of the at least one electrical circuit component, Γrepresents a Gamma function, and T represents the sampling period. 27.An apparatus in accordance with claim 15, wherein each of the at leastone electrical circuit component is arranged to exhibit double-layercharacteristics.
 28. An apparatus in accordance with claim 15, whereineach of the at least one electrical circuit component includes a batteryor a supercapacitor.